Tutorial for Sample Spaces and Events

Section 6.1: Sample Spaces and Events

(Based on Section 6.1 inFinite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World. )

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First, here are some basic definitions.

Definition Example

An experiment is an occurrence we observe whose result is uncertain. Throw a pair of dice and then add the numbers facing up.

An outcome is some specific aspect of the experiment that we observe. Any number from 2 to 12; for example, the following picture represents the outcome 7:

The sample space for the experiment is the set of all possible outcomes. The set of all numbers from 2 to 12:
S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

On-Line Dice Simulation If your browser is Java-capable, press the Java button to bring up an applet that simulates the above experiment.

In an experiment where a pair of dice (one red, one green) is thrown and the number facing up on each die is noted, the sample space is:

{1, 2, 3, 4, 5, 6}

{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

(1,1)	(1,2)	(1,3)	(1,4)	(1,5)	(1,6)
(2,1)	(2,2)	(2,3)	(2,4)	(2,5)	(2,6)
(3,1)	(3,2)	(3,3)	(3,4)	(3,5)	(3,6)
(4,1)	(4,2)	(4,3)	(4,4)	(4,5)	(4,6)
(5,1)	(5,2)	(5,3)	(5,4)	(5,5)	(5,6)
(6,1)	(6,2)	(6,3)	(6,4)	(6,5)	(6,6)

A coin is tossed three times in succession, and the total number of times heads comes up is noted. The sample space is:

	{0, 1, 2, 3}			{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
	{HHH, HHT, HTH, HTT, THH, THT, TTH}

You can consult Example 2 on p. 391 inFinite Mathematics Applied to the Real World or Finite Mathematics and Calculus Applied to the Real World for additional practice.

Event
Given a sample space S, an event E is a subset of S. The outcomes in E are called the favorable outcomes. We say that E occurs in a particular experiment if the outcome of that experiment is one of the elements of E, that is, if the outcome of the experiment is favorable.
How To Determine The Set E: Just say this to yourself when you are looking for the event E:
The event E consists of all outcomes in S which are favorable.

Example
Imogen enjoys sitting in front of the TV and randomly grabbing two chocolates at a time from her snack jar. The snack jar contains a large number of nut log swirls, turkish delights, and mocha surprises. Describe the sample space, and also the event that Imogen selects at least one mocha surprise in her first grab.

Solution
Here, the sample space consists of a single pair of chocolates (we are only interested in the first time she grabs two chocolates. We can therefore take the set of outcomes (sample space) to be

S = {NN, NT, NM, TN, TT, TM, MN, MT, MM},

where N = nut log swirl, T = turkish delight, and M = mocha surprise.

Now for the event E. Using the above suggestion, write down the following:

The event E consists of all outcomes in S which are favorable.

Since the favorable outcomes are those with at least one mocha surprise, we have

The event E consists of all outomes in S which contain at least one moca surprise.

Thus,

E = {NM, TM, MN, MT, MM}.

(Just delete those outcomes containing no M.)

The next example is based on Example 3 on p. 392 inFinite Mathematics Applied to the Real World or Finite Mathematics and Calculus Applied to the Real World.

A factory worker may be covered by employer's group plan, covered by his or her own individual plan, covered by the spouse's individual plan, or not covered at all. The event that a worker is not covered by an individual plan is:

	{not covered by an individual plan}			two out of four
	{covered by employer's plan, not covered}			{covered by own plan, covered by spouse's plan}

A coin is tossed three times, and the sequence of heads and/or tails is noted. The event that heads comes up at least twice is:

	{2, 3}			two out of four
	{HHH, HHT, HTH, THH}			{HHT, HHH}

Operations on Events

Set Operation Example

The complement, E', of an event E is the event that E does not occur. It is the set of all outcomes not in E. Toss a pair of dice and then add the numbers facing up. If E is the event that the sum is even, then E' is the event that the sum is odd:
S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
E = {2, 4, 6, 8, 10, 12}
E' = {3, 5, 7, 9, 11}

The union, EF, of events E and F is the event that either E occurs or F occurs (or both). Toss three coins and record the sequence of heads and tails. If E is the event that heads come up only once, and F is the event that tails come up only once, then EF is the event that either heads come up only once, or tails come up only once.
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
E = {HTT, THT, TTH}, F = { HHT, HTH, THH}
EF = {HTT, THT, TTH, HHT, HTH, THH}

The interectsion, EF, of events E and F is the event that noth E and F occur. Pick a three-digit number (000-999) at random. If E is the event that the first digit is 9, and F is the event that the remaining digits add up to 2, then EF is the event that the first digit is 9 and the remaining digits add to 2.
S = set of all three-digit numbers (1,000 of them)!
E = the set of all numbers 900 through 999
F = the set of all numbers of the form *02, *11, or *20
EF = {902, 911, 920}
Press the little jewel to see the the whole sample space and the events in all their gory detail!

If E and F are events then E and F are said to be disjoint or mutually exclusive if EF is empty. In the experiment immediately above, take E to be the event that the first digit is 9 and F to be the event that the first digit is 8. Then E and F are mutually exclusive.

(Similar to Example 5 on p. 395 of Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World )

In the experiment where a pair of dice (one red, one green) is thrown and the number facing up on each die is noted, let E be the event that the sum of the numbers is 4, and let F be the event that the sum is an odd number. The event F' is:

	the event that the outcome is an even number			{(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)}
	{2, 4, 6, 8, 10}			the event that the outcome is an even number other than 4

With E and F as described above, EF' is the event

	{(1, 1), (1, 5), (2, 4), (2, 6), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)}			{(1, 3), (2, 2), (3, 1)}
	same as E			the empty set

With E and F as described above, E'F is the event

	that the sum of the numbers is an odd number other than 4			that the sum of the numbers is any number
	that the sum of the numbers is any even number other than 4			that the sum of the numbers is any number other than 4

For more practice, try some of the questions in the true/false quiz (warning: it covers the whole of chapter 6) by pressing the button on the sidebar. Then try the exercises on pp. 399-404 of Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.

Definition	Example
An experiment is an occurrence we observe whose result is uncertain.	Throw a pair of dice and then add the numbers facing up.
An outcome is some specific aspect of the experiment that we observe.	Any number from 2 to 12; for example, the following picture represents the outcome 7:
The sample space for the experiment is the set of all possible outcomes.	The set of all numbers from 2 to 12: S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Set Operation	Example
The complement, E', of an event E is the event that E does not occur. It is the set of all outcomes not in E.	Toss a pair of dice and then add the numbers facing up. If E is the event that the sum is even, then E' is the event that the sum is odd: S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} E = {2, 4, 6, 8, 10, 12} E' = {3, 5, 7, 9, 11}
The union, EF, of events E and F is the event that either E occurs or F occurs (or both).	Toss three coins and record the sequence of heads and tails. If E is the event that heads come up only once, and F is the event that tails come up only once, then EF is the event that either heads come up only once, or tails come up only once. S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} E = {HTT, THT, TTH}, F = { HHT, HTH, THH} EF = {HTT, THT, TTH, HHT, HTH, THH}
The interectsion, EF, of events E and F is the event that noth E and F occur.	Pick a three-digit number (000-999) at random. If E is the event that the first digit is 9, and F is the event that the remaining digits add up to 2, then EF is the event that the first digit is 9 and the remaining digits add to 2. S = set of all three-digit numbers (1,000 of them)! E = the set of all numbers 900 through 999 F = the set of all numbers of the form 02, 11, or *20 EF = {902, 911, 920} Press the little jewel to see the the whole sample space and the events in all their gory detail!
If E and F are events then E and F are said to be disjoint or mutually exclusive if EF is empty.	In the experiment immediately above, take E to be the event that the first digit is 9 and F to be the event that the first digit is 8. Then E and F are mutually exclusive.